(1) I just think calling a nonrandomized study a correlational study is weird.
(2) I meant to say effect; not study; fixed
(3) If something is caused by a confounding variable, then the independent variable may have no relationship with the dependent variable. You seem to be using correlation to mean the result of an analysis, but I'm thinking of it as the actual real relationship which is distinct from causation. So y=x does not mean y causes x or that x causes y.
I don't understand what you mean by "real relationship". I suggest tabooing the terms "real relationship" and "no relationship".
I am using the word "correlation" to discuss whether the observed variable X predicts the observed variable Y in the (hypothetical?) superpopulation from which the sample was drawn. Such a correlation can exist even if neither variable causes the other.
If X predicts Y in the superpopulation (regardless of causality), the correlation will indeed be real. The only possible definition I can t...
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